Characterizing topology at nonzero temperature

TL;DR

This work investigates how to diagnose topology in one-dimensional inversion-symmetric fermionic chains at nonzero temperature, focusing on the SSH model with nearest and next-nearest neighbor hopping. It evaluates three approaches: (i) the ensemble geometric phase (EGP), (ii) local density operators acting on adjacent sites, and (iii) a mixed-state generalization of the local chiral marker via a purity-gap band-flattened projector $P$. The main findings are that the EGP phase remains well defined and reduces to the zero-temperature Zak phase in the thermodynamic limit, but the modulus $|\langle T\rangle|$ vanishes exponentially with system size, limiting practicality; meanwhile, the local densities and the chiral marker provide robust, local real-space diagnostics that distinguish three topological sectors when $t_3$ is present and remain meaningful at finite $T$ as long as a purity gap persists. Together these results offer complementary, scalable tools for identifying and measuring finite-temperature topology in mixed Gaussian states.

Abstract

We compare different methods for characterizing topology at nonzero temperatures, as applied to one-dimensional inversion-symmetric fermionic chains and focusing on the Su-Schrieffer-Heeger model with nearest- and next-nearest-neighbor hopping. Whilst the ensemble geometric phase, a mixed-state generalization of the Zak phase remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop a diagnostic that is suitable also for very large systems, we introduce local density operators acting on neighboring sites, which distinguish topological phases by comparing their expectation values. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be adiabatically flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real space topological invariant that coincides with the winding number in the zero temperature limit. The ensemble geometric phase, the local density operators, and the local chiral marker, provide complementary schemes to identify and measure topological phases of the Su-Schrieffer-Heeger chain beyond pure states.

Figures (6)

• Figure 1: The SSH model with unit cells labeled by $j$, each containing two sites $A$ and $B$. $t_1$ (solid), $t_2$ (double solid), and $t_3$ (dotted) denote the intracell, intercell, and next-nearest-neighbor hoppings, respectively.
• Figure 2: The amplitude $|\langle T \rangle|$ as a function of $t_1/t_2$ and inverse temperature $\beta$, where $t_1$ is the intracell hopping energy, $t_2=2$ is the intercell hopping energy in the SSH model. The number of unit cells is $N=150$. The overlaid contour lines in brown indicate the loci where $|\langle T \rangle|=0.01$ for systems with $N=150$ (solid), $N=100$ (dashed), and $N=50$ (dotted).
• Figure 3: The SSH chains with three cells labeled by $j$, depicting the local intracell operator $T_{j-1}^{\rm{intra}}$, the local inter cell operator $T_{j}^{\rm{inter}}$, and the nonlocal next-nearest-neighbor operator $T_{j}^{\rm{nnn}}$.
• Figure 4: (a): The magnitude of the expectation value of the local density operator, $T_j^{\ell}$, where $\ell$ corresponds to either intra or inter, as a function of cell position $j$ for three values of inverse temperature $\beta$, and for the two phases where $t_1=2, t_2=1$ colored in green, and $t_1=1, t_2=2$ colored in pink. $\ell=$intra is depicted with a solid line, and $\ell=$inter is depicted with a dashed line. Square markers correspond to $\beta=\infty$ (zero temperature), and circular markers to $\beta=1$. (b): The minimum value of the magnitude of the expectation values of the intra (pink) and inter (green) cell operators as a function of $t_1/t_2$ where $t_2=2$, for $N=19$. The square, circular, and triangular markers correspond to $\beta=\infty$, $\beta=3$, and $\beta=1$ respectively. (c): $\mathcal{T}$ as a function of $\beta$ and $t_1/t_2$, where $t_2=2$, and $N=19$.
• Figure 5: (a): $\mathcal{T}_{3}$ as a function of the ratios $t_3/t_1$ and $t_2/t_1$,the hopping parameters of the next-nearest-neighbor SSH model, at zero temperature. The dotted line denotes $\mathcal{T}_{3}=0$. (b): $\mathcal{I}$, as a function of $t_3/t_1$ and $t_2/t_1$ at zero temperature, where the difference in sign distinguishes between two topological phases. (c): $\mathcal{T}_{3}$ as a function of inverse temperature $\beta$ and $t_2/t_1$. (d): $\mathcal{I}$ as a function of inverse temperature $\beta$ and $t_2/t_1$. $t_1=2$ in all panels, and $t_3=1.5$ in panels (c) and (d). The number of unit cells is $N=32$ in all four panels.